The MMD requires the use of characteristic kernels where a kernel is characteristic if \(\text{MMD}(\mathcal {F}, \mathcal {D}^{\prime }, \mathcal {D}) = 0 \Leftrightarrow \mathcal {D}^{\prime } = \mathcal {D}\)  [1]}. Without any assumption, it is not straightforward to determine if an arbitrary kernel is characteristic. However, [2]} have shown that a stationary kernel is characteristic if it can be written in the form \(\psi (x-x’)\)  where \(\psi \)  is a bounded, continuous, real-valued, positive definite function on \(\mathbb {R}^d\)  with a compact support, which is relatively straightforward to check. This motivates the use of stationary kernels with the MMD metric. The stationary kernels include the class of isotropic kernels whose \(k(x, x’)\)  only depends on the distance between data points \(\Vert x-x’\Vert \) . The squared exponential and Laplace kernels are examples of isotropic kernels shown by [3]} to be characteristic.
